CMO 2013

1. Two circles and of different radii intersect at two points and , let and be two points on and , respectively, such that is the midpoint of the segment . The extension of meets at another point , the extension of meets at another point . Let and be the perpendicular bisectors of and , respectively. Show that and have a unique common point (denoted by ). Prove that the lengths of , and are the side lengths of a right triangle.

2. Find all nonempty sets of integers such that for all (not necessarily distinct) .

3. Find all positive real numbers with the following property: there exists an infinite set of real numbers such that the inequality

holds for all (not necessarily distinct) , all real numbers and all positive real numbers .

4. , there are finite sets satisfy that for any , . Try to find the the minimum of .

5. For any positive integer and , denote , where , define . are positive integers, and for any . Prove that if , then for any positive integer .

6. are positive integers, find the minimum positive integer satisties that if is a set of integers contains a complete residue system of and , then there exit a nonempty set , that .